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I have that problem, too. I use "shoe-goo" shoe repair and it works pretty well. It usually lasts for several months before I have to do it again. It occurs to me that alternating layers of Flex Seal and good quality cloth gaffer's tape would make a very strong, flexible, watertight patch as well.
You can also do it the old fashioned way by putting the wood in a metal container with a pinhole in it and placing the container in a fire for 10 minutes or so. For such a small piece, you could probably use an altoids tin or an old metal 35mm film can and a blowtourch. https://www.instructables.com/id/Make-Your-Own-Charcoal-at-Home-Video/
Excellent and elegant. I'll definitely be doing this!
It seems like you'd get smoother, more accurate results if the in-feed side hole is larger than the desired out-feed size. Adjust the blade height to exactly line up with the bottom edge of the out-feed hole. Taper the tip a bit to start it and it should feed through nice and smoothly, with very little vibration on the other side since the exiting dowel very closely matches the guide hole. You might even pass it through a block of foam to be sure to dampen any vibration. At least, it works in my head. :) I'll have to give it a try tomorrow. Thanks for sharing!